Solve the trigonometric series equation $1+\mathrm{sin}x+{\mathrm{sin}}^{2}x+{\mathrm{sin}}^{3}x+\cdots =4+2\sqrt{3}$

Paul Duran
2022-05-07
Answered

Solve the trigonometric series equation $1+\mathrm{sin}x+{\mathrm{sin}}^{2}x+{\mathrm{sin}}^{3}x+\cdots =4+2\sqrt{3}$

You can still ask an expert for help

Leroy Lowery

Answered 2022-05-08
Author has **22** answers

$\mathrm{\forall}x\in \mathbb{R}|\mathrm{sin}x|\le 1$ and if $|u|<1$ then $1+u+{u}^{2}+{u}^{3}+...=\frac{1}{1-u}$

$\frac{1}{1-\mathrm{sin}x}=4+2\sqrt{3}$

$\mathrm{sin}x=\frac{\sqrt{3}}{2}$

$x=(-1{)}^{k}\frac{\pi}{3}+\pi k,k\in \mathbb{Z}$

$\frac{1}{1-\mathrm{sin}x}=4+2\sqrt{3}$

$\mathrm{sin}x=\frac{\sqrt{3}}{2}$

$x=(-1{)}^{k}\frac{\pi}{3}+\pi k,k\in \mathbb{Z}$

asked 2021-06-16

Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 11 passengers per minute.

asked 2020-10-18

If $\mathrm{sin}x+\mathrm{sin}y=a{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\mathrm{cos}x+\mathrm{cos}y=b$ then find $\mathrm{tan}(x-\frac{y}{2})$

asked 2022-01-27

Find the set of values, which are taken by $\mathrm{tan}z$

I was trying to write tangents as follows:

$\mathrm{tan}\left(z\right)=-i\frac{{e}^{iz}-{e}^{-iz}}{{e}^{iz}+{e}^{-iz}}$

and then

z=a+bi

, which led me to

$\mathrm{tan}z=-i\frac{\mathrm{cos}a({e}^{-b}-{e}^{b})+i\mathrm{sin}a({e}^{-b}+{e}^{b})}{\mathrm{cos}a({e}^{-b}+{e}^{b})+i\mathrm{sin}a({e}^{-b}-{e}^{b})}$

I was trying to write tangents as follows:

and then

z=a+bi

, which led me to

asked 2021-09-05

Proof trigonometry identities.

$\mathrm{cot}\left(x\right)+\mathrm{tan}\left(x\right)+\mathrm{sec}\left(x\right)\mathrm{csc}\left(x\right)=2\mathrm{sec}\left(x\right)\mathrm{csc}\left(x\right)$

asked 2022-01-03

How can I prove the identity

$\frac{1}{1+\mathrm{sin}\left(x\right)}\equiv \frac{{\mathrm{sec}}^{2}\left(\frac{x}{2}\right)}{{(\mathrm{tan}\left(\frac{x}{2}\right)+1)}^{2}}$

asked 2022-01-16

Is my evaluation of complex trigonometric expression $\mathrm{sin}(\frac{\pi}{4}+2i)-\mathrm{sin}\left(2i\right)$ correct?

asked 2022-04-15

Find $\mathrm{sin}\frac{\theta}{2}$ when $\mathrm{sin}\theta =\frac{35}{}$ , and $0}^{\circ}<\theta <{90}^{\circ$

The answer I'm currently getting is$\sqrt{\frac{1}{10}}$ but the answer must be either $\frac{\sqrt{10}}{10},\frac{3\sqrt{10}}{10}\text{}\text{or}\text{}\frac{13}{}$

My Process:

since$\mathrm{sin}\frac{\theta}{2}=\pm \sqrt{\frac{1-\mathrm{cos}\theta}{2}}$

$\sqrt{\frac{1-\frac{4}{5}}{2}}\cdot \sqrt{\frac{5}{5}}=\sqrt{\frac{5-4}{10}}=\sqrt{\frac{1}{10}}$

The answer I'm currently getting is

My Process:

since